A Historical Introduction to the
Philosophy of Science

The following is a summary of the sixteenth chapter of John Losee's book, A Historical Introduction to the Philosophy of Science (fourth edition), with some ancillary notes.

Philosophers discussed in this chapter: Clark Glymour (1942—); Thomas Kuhn (1922–96); Imre Lakatos (1922–74)

Bayesian Confirmation Theory

(p. 220) Goodman's 'New Riddle of Induction' based on his 'grue' paradox forced some philosophers to abandon the Logical Reconstructionist's syntactical theory of confirmation for a quantitative probability theory.

(p. 221) The axioms of the probability calculus entail:

1. The probability for a sentence (A) being true is greater than or equal to 0.
2. The probability for a tautology (t) (logical truth) being true is 1 (certainty).
3. The probability for either of two inconsistent sentences (A, B) being true equals the addition of the probabilities of the individual sentences being true (P(A) + P(B)).
4. The probability for a sentence (A) being true given another sentence (B) being true equals the probability of both sentences being true (P(A) + P(B)) divided by the probability of the latter sentence (B) being true.

1. to 4. entail Bayes' Theorem:

The probability for a sentence (A) being true given another sentence (B) being true equals the product (X) of
the probability for sentence (B) being true given sentence (A) being true and the probability for sentence (A) being true, divided by
the probability for sentence (B) being true.

Bayes' Theorem is adapted by some to give a quantitative measure of evidential support for a scientific hypothesis (h) given an observational evidence statement (e):

The probability for a hypothesis (h) being true given evidence statement (e) being true equals the product (X) of
the probability for evidence statement (e) being true given hypothesis (h) being true and the probability hypothesis (h) being true, divided by
the probability for evidence statement (e) being true.

The probability for evidence statement (e) being true is the sum (+) of the individual probabilities of the evidence statement (e) being true given each of the hypotheses competing with hypothesis (h).

Bayes' Theorem accords with our intuition that the degree of confirmation given by evidence statement (e) to a hypothesis (h) equals

the probability for hypothesis (h) being true given evidence statement (e) being true minus (–) the initial probability of hypothesis (h) being true

(p. 222) and that the degree of confirmation:

• increases the more probable that evidence statement (e) is true given hypothesis (h) is true; and
• decreases the more probable that one of the hypotheses competing with hypothesis (h) is true.

Given this relation:

The probability of a hypothesis being true given an evidence statement (e) being true relative to a competing hypothesis being true given the same evidence statement (e) being true
is equal to
the product (X) of the probability for evidence statement (e) being true given hypothesis (h) being true and the probability hypothesis (h) being true
relative to
the product (X) of the probability for evidence statement (e) being true given the competing hypothesis (h*) being true and the probability for hypothesis (h*) being true.

For Bayesians to solve the 'grue' paradox, they need to accord a higher prior probability to the 'green hypothesis'.
[LA: Or find an evidence statement entailed by the 'green' hypothesis that is not entailed by the 'grue' hypothesis, or vice versa; e.g., find a time-variant colour mechanism in emeralds.]

Three ways for Bayesians to interpret 'the probability of a hypothesis':

1. 'frequency interpretation' – frequencies of occurrence in a long-run series of trials
2. 'logical interpretation' – logical relations between hypotheses and evidence statements
3. 'subjectivist interpretation' – measures of rational belief

Losee thinks Bayesians' subjectivist interpretation poses the problem of how to assign degrees of rational belief to a hypothesis. Bayesians respond that scientists applying Bayes' Theorem to new evidence converge in their assessment of posterior probability (as with assessing the probability of drawing a white ball at random from an urn).
[LA: For a suggestion for a satisfactory interpretation of 'prior probability', see my ''Bayes' Theorem and Theory Appraisal'.]

(p. 223) But Bayes' Theorem treats the repetition of the same experiment as important as varied and severe tests of a hypothesis (e.g., tests of the law of refraction). Bayesians respond that Bayes' Theorem is not designed to assess confidence in particular experimental results. [LA: But that's not the objection.]

Glymour complained that Bayesians have only offered a theory of personal learning, not a theory of scientific reasoning.

(p. 224) Glymour also objected that for positive evidence already known at the time of the formulation of a hypothesis, Bayes' Theorem counts it as of zero value. Bayesians Howson and Urbach responded that Bayes' Theorem should be applied counterfactually; as if the evidence was newly discovered.

Bayesian Garber suggested that prior known evidence adds support for the hypothesis because it becomes known that the hypothesis entails the evidence statement. Losee objects that a hypothesis does not entail the evidence statement alone.

(p. 225) For Gerber, support for a theory is of two types:

1. new evidence raises theory's posterior probability
2. new discovery that theory entails prior known evidence

Miller noted that in the face of new counter-evidence, a scientist may adjust the prior probabilities to save their favoured hypothesis (e.g., creationist ad hoc adjustment).

Bayesians fatally lack a rule for when to disallow such ad hoc revisions. Many such revisions were legitimate in the history of science (e.g., Darwin on the fossil record, Copernicus on stellar parallax, Galileo on telescopic magnification).

Glymour on "Bootstrapping"

(p. 226) Glymour pointed out how a scientific theory may receive supporting evidence from another part of the same theory ('bootstrapping') (e.g., Newton's Second Law supported his universal law of gravitation via the motions of Jupiter's moons).

(p. 227) Contra historical theories of confirmation, Glymour's Bootstrap Model is in the Logical Reconstructionist semantic-relations tradition that sees the time that evidence is discovered as irrelevant to the support it gives to a theory.

Lakatos on Comparative Confirmation

For Lakatos, prior known evidence adds support for a hypothesis when:

1. the hypothesis (plus relevant conditions plus auxiliary hypotheses) entails the prior known evidence; and

2. either the main rival hypothesis (plus relevant conditions plus auxiliary hypotheses):

   1. entails the negation of the prior known evidence; or
   2. entails neither the prior known evidence nor the negation of the prior known evidence

An example is how old experiments showing weight gain on combustion confirmed Lavoisier's Oxygen Theory as they contradicted the rival Phlogiston Theory.

Theory Appraisal

(p. 228) Duhem and Campbell showed how the deducibility of lower-level laws does not confirm a theory. As a solution, Kuhn offered the following prescriptive and historically descriptive criteria for theory acceptance:

1. internal consistency
2. agreement with observations
3. simplicity
4. breadth of scope
5. conceptual integration
6. fertility

Losee agrees that criterion 1. (consistency) is a necessary condition for theory acceptance.

(p. 229) Losee argues that the deducibility of an observation report (criterion 2.) is debated by scientists in specific cases.

Similarly, criterion 3. (simplicity) is also vague (e.g., Is power of the independent variable or the number of variables the criteria?)

Kuhn recognized that his criteria conflict in certain cases (e.g., agreement with observations conflicts with simplicity of relationship between variables).

Kuhn's criterion 4. (breadth of scope) is well-supported by the historical record (e.g., Newtonian Mechanics, Electromagnetic Theory of Light).

(p. 230) Kuhn's criterion 5. (conceptual integration) is also well-supported by the historical record (e.g., Copernicus' dealing with retrograde motion).

So is Kuhn's criterion 6. (fertility). McMullin identified two types of fertility of a theory:

1. 'proven fertility' – successful adaptation to historical pressures (expanding explanatory power, resolving anomalies)
2. 'potential fertility' – successful adaptation to future pressures

A theory may be fertile either by:

1. showing how to make progressive modifications to the theory itself (e.g., Bohr's atomic theory); or
2. applying itself successfully to a new type of phenomena (e.g., LaPlace's theory of heat) (p. 231)

(p. 231) Debates about the fertility of LaPlace's theory of heat may indicate that 'application to new phenomena' is a function of scientists' surprise.

Zahar argued for an objective basis for 'novel facts'. For Zahar, a fact is a 'novel fact' if the theory under appraisal was constructed to solve problems that did not include that fact.

Novel facts supporting a theory are of two kinds:

1. novel prediction – only known to be true after theory formulation (e.g., Mendeleeff's predictions of new elements; Maxwell's prediction of gas viscosity relation)
2. novel postdiction – known to be true before theory formulation, but not part of theory formulation (e.g., LaPlace's deduction of discrepancy between calculated and actual sound velocities)

(p. 232) Zahar also argued the Michelson–Morley null result over the speed of light in an hypothesized ether as an example of 'novel' support for Einstein's Special Relativity Theory.

Losee insists we keep separate:

1. truth of predictivist thesis – a newly known fact provides greater support to a theory compared with a fact known prior and accommodated during theory construction
2. problem of undesigned scope – whether novel deducibility is measurable and ought to be a criterion of theory acceptance

Brush's historical research shows that scientists have not accepted a theory primarily on the basis of novel facts.

(pp. 232–3) Kuhn insisted that scientists apply his several criteria for theory choice, although there is no set algorithmic calculation. This is because with scientists' varied and idiosyncratic personalities, they disagree on how to weigh the criteria when they conflict.

McAllister on Aesthetic Standards

(p. 233) Contra Kuhn's 'idiosyncratic factors' determining scientists' theory appraisal, McAllister advocated a 'rationalist' approach that first accepted Kuhn's inviolable 'logico-empirical' criteria:

• internal consistency
• agreement with data
• novel prediction

But McAllister further added aesthetic standards of theory choice, revisable by scientists during periods of scientific revolution:

• visualizability
• symmetry
• explanatory simplicity
• ontological parsimony

(pp. 233–4) For McAllister, Kepler, Bohr and Heisenberg were revolutionary in rejecting the current aesthetic standards. But Copernicus was not as he accepted Platonic–Aristotelian uniform circular motion. Likewise, Einstein was not as he accepted the aesthetic symmetry considerations of the day.

(p. 234) Losee objects that even if we accept McAllister's criteria for revolutionary theory change, it's uncertain what counts as an 'aesthetic standard' (e.g., Copernicus may be counted as overturning the Platonic–Aristotelian aesthetic of the crystalline sphere).

Further, de Regt noted that Einstein, with his Special Relativity Theory, rejected the 'aesthetic' standard of absolute containers. McAllister's 'aesthetic' standard is too loose a criterion for theory choice.

Questions to Consider:

1. Of the three ways of interpreting 'the probability of a hypothesis', which do you think is the most promising for Bayes' Theorists?
2. Is Gerber's modification of the application of Bayes' Theorem legitimate?
3. Are Kuhn's criteria for theory appraisal sufficiently precise to act as a useful guide for scientists?
4. Has Zahar accounted adequately for how old facts can objectively support a new theory?
5. How much do think aesthetic standards play in scientists' evaluation of alternative theories?

Copyright © 2022–3 Leslie Allan